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That said, I agree with @Chuck's suggestion that it might be useful to have a canonical sceptics-style question about the effectiveness of brain training.

Questions

What is the effect of completing "brain training"?

Is there any evidence for domain general benefits to cognitive functioning that extend beyond the specific task practiced?

Is there anything particularly "neural" or "brain"-like about brain training?

Is there any evidence that brain training prevents any neurological diseases?

Naturally, it would be good if any answers are backed up by independent scientific empirical research. References to and summaries of recent scientific review articles would also be particularly useful.

Numerous studies have shown that mental stimulation is effective in reducing the risk of Alzheimer's disease. The more interesting question is probably whether brain training can ameliorate attention deficit...

Is there any evidence for domain general benefits to cognitive functioning that extend beyond the specific task practiced?

Brain training at the very least improves skill levels in the domain being trained. That is now well established. The big challenge is of course to create forms of training whose effects generalize beyond the domain trained. Things have not been so promising to date in this regard. According to one recent study by Adrian Owen (Owen, A., et al., 2010, "Putting brain training to the test". Nature 465: 775–779) training effects barely generalize to even near similar domains. The weight of evidence is that as of yet no popular brain training methods can significantly increase mental ability in any domain. The only exciting exceptions are the well established effects of the dual n-back task on fluid intelligence (a subtype of intelligence measured using the Raven's matrices test), and the newly emerging relational frame training approach which seems to lead to large increases in full scale IQ, despite the fact that the training does not resemble IQ test content.

Is there any evidence that brain training prevents any neurological diseases?

Brain training critics are only too familiar with the ACTIVE studies that have all but dismissed the effects of brain training methods on cognitive decline in the elderly (e.g., Ball, K; Berch DB; Helmers KF; et al., 2002. "Effects of cognitive training interventions with older adults: A randomized controlled trial". JAMA 288(18): 2271–81. doi:10.1001/jama.288.18.2271. PMC 2916176. PMID 12425704). The public confusion around this issue stems from the fact that is it is not very well established that keeping our brains and minds active does protect against cognitive decline. The problem is that it is not yet known if brain training alone is sufficient to prevent decline in cognitive abilities as we age. So researchers are not incorrect that brain stimulation may well stem cognitive decline, but it does not follow from this that brain training can produce the same effects as perhaps decades of mental and physical stimulation across a wide variety of contexts. The theory certainly suggest that brain training should be able to have these positive effects, but it is too early to say for sure yet if we have developed techniques that can
reliably do this.

Is there anything particularly "neural" or "brain"-like about brain training?

Everything we do "trains" our brains. We know this from decades of research on neuroplasticity or "neurogenesis". The problem, however, is that not all forms of mental stimulation lead to neural changes that allow us to perform new skills more easily. If we practice the piano, we get better at the piano, and in so doing our brain changes measurably. But the promise of brain training is that practicing a skill such as piano playing might make us better at math for example. This kind of generalization of skills across domains is not so easy to prove, even though it is now well established that neurogenesis does occur.

The general public is easily fooled that any training that changes our brain and builds grey matter must be good, but fails to understand that this is often of little consequence. More recent behavioral approaches, in the field of Relational Frame Theory are starting to show that generalization can be achieved across skill domains. Using a technique called relational training, researchers are starting to see changes in verbal ability, from training in forms of logical reasoning for example.